My first suggestion is hit the exchange and upgrade your equipment. Look for what you need and check the numbers, not the colour. Many times you'll have better numbers with white conned level appropriate gear than you will with lesser leveled green or blues. Get everything on your ship to mk IV. Make sure your bridge officers have the proper skills... shield heals for sci, hull heals for eng, and damage buffs appropriate for your weapon type... cannons vs. beams. (Avoid Beam Overload as it's still broken... or was last week before I dropped it.)

I had this same problem when I first started. I'd get a new ship, go flying into combat and get my arse handed to me. It took me a day or two to realize that new ships come outfitted with equipment 1 level lower than you are. (and longer to figure out that the numbers on some stuff just don't add up. i.e. a blue con mk iii whatchamacallit that has worse stats than a white con mk iv and sometimes mk iii.)

You really don't need 'good' gear to do most of the content. White con 'vendor trash' will get you through until you can replace it with a mission reward or drop.